Is it possible to add widgets to one page and not have them on the main site?

Yes.

Can I stack widgets on the page, assuming that the whole page is set up for widget use?

Yes.

Is there a theme where the whole page is widgetized?

I don't know off hand, but probably.

Is it easy to alter a theme to let an area have a widget added?

Yes.

Is it possible to make the page theme completely different from the rest of the site? How would I do that?

Yes. Create a new template and edit the PHP/html and the CSS associated with it to get whatever look you want. Then add a new page via Pages->Add New using your new template. Then set that page as the Home page via Setting->Reading Static home page.